Theoretical derivation of dimensionless length scale. As discussed in the pioneering works [1]}, [2]},
the rheology of the granular flow is represented by the microscopic particle movement time scale \(t_{f}\)  divided by the macroscopic rearrangement time scale \(\mathcal {T}=1/\dot{\gamma }\) . \(\mathcal {T}\)  is the time for one layer of particles to be sheared over one particle diameter, \(d\) , with respect to the other, and \(t_{f}\)  can be seen as the time for a particle to be compressed by the confining pressure \(P\)  to travel over a particle diameter. Following the work of Cassar et al.[2]}, the characteristic equilibrium of a single particle inside an immersed granular assembly is given by
\((\pi /6)\rho _{s}d^3\frac{d u}{d t}=(\pi /4) P d^2-F_{d},\) 
